LESSON 25: LAGRANGE MULTIPLIERS OCTOBER 30, 2017
|
|
- Gilbert Hudson
- 6 years ago
- Views:
Transcription
1 LESSON 5: LAGRANGE MULTIPLIERS OCTOBER 30, 017 Lagrange multipliers is another method of finding minima and maxima of functions of more than one variable. In fact, many of the problems from the last homework would have been easier via this method. However, this method only applies if you need to find the extrema subject to some constraint. So not every homework problem could have been done using this method. The Method of Lagrange Multipliers: Suppose we want to minimize or maximize a function f(x, y) subject to the constraint g(x, y) = C. Introduce a dummy variable, λ, and solve the system of equations (1) () (3) for (x, y). f x (x, y) = λg x (x, y) f y (x, y) = λg y (x, y) g(x, y) = C Ex 1. Maximize the area of a rectangular garden subject to the constraint that its perimeter is 100 ft. Solution: Let x be the length and y the width of the garden. Then the function we are maximizing is But this is subject to the constraint that f(x, y) = xy. x + y = 100. perimeter By our method, we set up a system of equations We solve for x and y. y = λ () f x g x x = λ () f y x + y g(x,y) g y = 100 C = λ = λ Since x = λ and y = λ, we see that given x + y = 100, 100 = ( λ ) + ( λ ) = 4λ + 4λ = 8λ. x y 1
2 MATH 1600 Then λ = = 5. Therefore, x = ( ) ( ) 5 5 = 5 and y = = 5. We conclude that the area is maximized when x = 5 and y = 5 and the maximum area of the garden is 5(5) = 65 ft. Question: How do we know this is a maximum and not a minimum? Because the minimum area of the garden is 0 ft. For these problems, you always need to consider whether your answer makes sense in context. Note 1. Lagrange multipliers will never tell you if there is a saddle point. Examples. 1. Minimize f(x, y) = (x + 1) + (y ) subject to g(x, y) = x + y = 15. Solution: Taking derivatives, we see that f x = (x + 1), f y = (y ), g x = x, g y = y. Setting up our equations (x + 1) = λ(x) = λx (y ) = λ(y) = λy x + y = 15 The method of Lagrange multipliers calls for a little creativity so try to be flexible when solving these types of problems. We focus on the first two equations. We have Thus, (x + 1) = λx x + 1 = λx (y ) = λy y = λy x + 1 = λx x λx + 1 = 0 x(1 λ) + 1 = 0 x(1 λ) = 1 y = λy y λy = 0 y(1 λ) = 0 y(1 λ) = From this we gather that x, y 0 else these equations can t be true. This means that we may divide by x and y. Hence, 1 x = 1 λ = y, but by cross-multiplication this becomes y = x y = x.
3 By our constraint, g(x, y) = x + y = 1, we see 15 = x + y = x + ( x) = x + 4x = 5x 5 = x MATH We conclude that x = ±5, which implies y = (±5) = 10. extrema points are (5, 10) and ( 5, 10). Evaluating f(x, y) at these points, Thus, our f(5, 10) = (5 + 1) + ( 10 ) = 6 + ( 1) = = 180 Max f( 5, 10) = ( 5 + 1) + (10 ) = ( 4) + (8) = = 80 Min Therefore, the minimum of our function is 80.. Find the minimum value of x e y subject to y + x = 6. Solution: Whatever function we are finding the extrema for is our f(x, y) and the constraint is g(x, y). Thus, we have Next, we find derivatives: f(x, y) = x e y and g(x, y) = y + x = 6. f x = xe y, f y = x ye y, g x =, g y = 4y. We set up our equations to get xe y = λ x ye y = 4λy y + x = 6 By the first equation, we see that xe y = λ xe y = λ. Substituting this into the second equation, we get yx e y = 4 (xe y ) y. λ We may divide both sides by e y because this is never 0. Thus, this equation becomes x y = xy. Subtracting y from both sides, this becomes yx xy = 0 y(x x) = 0.
4 4 MATH 1600 Hence, either y = 0 or x x = 0 x(x ) = 0 x = 0 or x =. We check all three of these cases: Case 1. y = 0 Case. x = 0 Case 3. x = If y = 0, then our constraint implies that Thus, one solution is (3, 0). 0 + x = 6 x = 3. If x = 0, then by our constraint: y + 0 = 6 y = 3 y = ± 3. So two of our solutions are (0, 3) and (0, 3). If x =, then y + () = 6 y = y = 1 y = ±1. This adds another two solutions: (, 1) and (, 1). Putting this all together, our solutions are (3, 0), (0, 3), (0, 3), (, 1), (, 1). Finally, we check the function values: f(3, 0) = (3) e (0) = 9(1) = 9 f(0, 3) = (0) e ( 3) = 0 min f(0, 3) = (0) e ( 3) = 0 min f(, 1) = () e (1) = 4e max f(, 1) = () e ( 1) = 4e max Therefore, the function s minimum value is Find the extrema of f(x, y) = e xy subject to 9x + 4y 7. Solution: This is a slightly different problem than what we have encountered so far. Here, our constraint is an inequality rather than an equality. Fortunately, this is not as daunting as this may appear. We break this problem into two parts: first, we find the critical points of f(x, y) which are contained in the region described by g(x, y) = 9x + 4y 7, and second, we apply the Lagrange multiplier method to f(x, y) subject to g(x, y) = 9x + 4y = 7.
5 The derivatives of f(x, y) are MATH f x = ye xy and f y = xe xy. Setting these equal to 0, we see that x = 0, y = 0 because e xy is never 0. Since the point (0, 0) satisfies g(x, y) = 9x + 4y 7, we include this in our list of solutions. Now, we assume that g(x, y) = 9x + 4y = 7 and proceed with the Lagrange multiplier method as we have been. The derivatives of g(x, y) are g x = 18x and g y = 8y. Setting up our system of equations, Next, we solve for (x, y). ye xy = 18λx xe xy = 8λy 9x + 4y = 7 We observe that if any of x, y, or λ are 0, then x = 0 and y = 0. But this case has already been covered, so we assume that x, y, λ 0. This means we can divide by x and y to get λ by itself. Write y 18x e xy = λ and x 8y e xy = λ. Thus, y 18x e xy = x 8y e xy. Because e xy is never zero, we may divide through on both sides to get Cross-multiplying, we get y 18x = x 8y. 8y = 18x y = 18 8 x = 9 4 x. Now, we return to our constraint and substitute for y, 7 = 9x + 4y ( ) 9 = 9x x y = 9x + 9x = 18x.
6 6 MATH 1600 Solving for x, we see that x = ±. Since y = 9 4 x, we get y = 9 y = ±3. Therefore, we get the following 4 solutions: (, 3), (, 3), (, 3), (, 3). We need to check the function values at each of our solutions: f(0, 0) = e (0)(0) = 1 f(, 3) = e ()(3) = e 6 min f(, 3) = e ()( 3) = e 6 max f(, 3) = e ( )(3) = e 6 max f(, 3) = e ( )( 3) = e 6 min is e 6. Thus, the minimum function value is e 6 and the maximum function value 4. Find the minimum value of f(x, y) = y x 4x subject to y = 8 x. Solution: Our f(x, y) = y x 4x but we need to determine our g(x, y). We are told our constraint is y = 8 x and, adding x to both sides, we have x + y = 8. Hence g(x, y) = x + y = 8. Next, we differentiate: Setting up our equations, f x = x 4, f y = y, g x =, g y = 1. x 4 = λ y = λ x + y = 8 We know immediately that y = λ, so, substituting into the first equation, we get Dividing both sides by 4, we get x 4 = (y) = 4y. λ y = 1 x 1. According to our constraint, 8 = x + y = x + ( 1 ) x 1 = x 1 x 1 = 3 x 1. y
7 MATH We solve 8 = 3 x 1 for x: 8 = 3 x 1 9 = 3 x 18 = 3x 6 = x Since x = 6, we have y = 1 (6) 1 = 3 1 = 4. Thus, our solution is (6, 4). Plugging this into the function, f(6, 4) = ( 4) (6) 4(6) = = 44. Note. We should check that this is actually a minimum as opposed to a maximum. To do this, we check any other point that satisfies x + y = 8, say (0, 8). If 44 is a minimum, then we must have 44 < f(0, 8) = 8 (0) 4(0) = 64. So we can rest easy knowing that this really is the minimum of the function subject to the given constraint. 5. Find the maximum value of f(x, y) = 3 x3/ y subject to x = 10 y. Solution: Again, we need to rearrange our constraint a little to determine our g(x, y). Adding y to both sides of x = 10 y, we get x + y = 10. Hence, Next, we differentiate: g(x, y) = x + y = 10. f x = x 1/ y, f y = 3 x3/, g x = 1, g y = 1. Now, we set up our equations: x 1/ y = λ 3 x3/ = λ x + y = 10 Since λ = x 1/ y and λ = 3 x3/, we can write x 1/ y = 3 x3/. Multiplying through by x 1/, we get xy = 3 x.
8 8 MATH 1600 Subtracting xy from both sides and regrouping, this becomes 0 = ( ) 3 x xy = x 3 x y. Hence, either x = 0 or 3 x y = 0 x = y. We check both cases. 3 Case 1. x = 0 If x = 0, our constraint implies that 0 + y = 10 y = 10. Hence, one solution is (0, 10). Case. 3 x = y If x = y, then 3 10 = x + y = x + 3 x = 5 3 x 10 = 5 3 x x = 6. Thus, since y = (6) = 4, one solution is (6, 4). 3 Putting this together, we have two solutions: (0, 10) and (6, 4). We check the function values at these points: f(0, 10) = 3 (0)3/ (10) = 0 min f(6, 4) = 3 (6)3/ (4) = 16 6 max Therefore, the maximum value is 16 6.
LESSON 23: EXTREMA OF FUNCTIONS OF 2 VARIABLES OCTOBER 25, 2017
LESSON : EXTREMA OF FUNCTIONS OF VARIABLES OCTOBER 5, 017 Just like with functions of a single variable, we want to find the minima (plural of minimum) and maxima (plural of maximum) of functions of several
More informationDEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 2018
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 208 Version A refers to the regular exam and Version B to the make-up. Version A. A particle
More informationSolutions to Homework 7
Solutions to Homework 7 Exercise #3 in section 5.2: A rectangular box is inscribed in a hemisphere of radius r. Find the dimensions of the box of maximum volume. Solution: The base of the rectangular box
More informationMath 10C - Fall Final Exam
Math 1C - Fall 217 - Final Exam Problem 1. Consider the function f(x, y) = 1 x 2 (y 1) 2. (i) Draw the level curve through the point P (1, 2). Find the gradient of f at the point P and draw the gradient
More information2015 Math Camp Calculus Exam Solution
015 Math Camp Calculus Exam Solution Problem 1: x = x x +5 4+5 = 9 = 3 1. lim We also accepted ±3, even though it is not according to the prevailing convention 1. x x 4 x+4 =. lim 4 4+4 = 4 0 = 4 0 = We
More informationMath 210, Final Exam, Fall 2010 Problem 1 Solution. v cosθ = u. v Since the magnitudes of the vectors are positive, the sign of the dot product will
Math, Final Exam, Fall Problem Solution. Let u,, and v,,3. (a) Is the angle between u and v acute, obtuse, or right? (b) Find an equation for the plane through (,,) containing u and v. Solution: (a) The
More informationLesson 26 MA Nick Egbert
Overview There is no new material for this lesson. Here, we explore a multitude of fun word problems that we can now do using the method of Lagrange multipliers from the previous lesson. Examples Example
More informationLagrange Multipliers
Lagrange Multipliers 3-15-2018 The method of Lagrange multipliers deals with the problem of finding the maxima and minima of a function subject to a side condition, or constraint. Example. Find graphically
More informationMath 2163, Practice Exam II, Solution
Math 63, Practice Exam II, Solution. (a) f =< f s, f t >=< s e t, s e t >, an v v = , so D v f(, ) =< ()e, e > =< 4, 4 > = 4. (b) f =< xy 3, 3x y 4y 3 > an v =< cos π, sin π >=, so
More informationSolutions to Homework 5
Solutions to Homework 5 1.7 Problem. f = x 2 y, y 2 x. If we solve f = 0, we can find two solutions (0, 0) and (1, 1). At (0, 0), f xx = 0, f yy = 0, f xy = and D = f xx f yy fxy 2 < 0, therefore (0, 0)
More informationMATH Midterm 1 Sample 4
1. (15 marks) (a) (4 marks) Given the function: f(x, y) = arcsin x 2 + y, find its first order partial derivatives at the point (0, 3). Simplify your answers. Solution: Compute the first order partial
More informationMultivariable Calculus and Matrix Algebra-Summer 2017
Multivariable Calculus and Matrix Algebra-Summer 017 Homework 4 Solutions Note that the solutions below are for the latest version of the problems posted. For those of you who worked on an earlier version
More informationMath 212-Lecture Interior critical points of functions of two variables
Math 212-Lecture 24 13.10. Interior critical points of functions of two variables Previously, we have concluded that if f has derivatives, all interior local min or local max should be critical points.
More informationMath 2142 Homework 5 Part 1 Solutions
Math 2142 Homework 5 Part 1 Solutions Problem 1. For the following homogeneous second order differential equations, give the general solution and the particular solution satisfying the given initial conditions.
More informationThe University of British Columbia Midterm 1 Solutions - February 3, 2012 Mathematics 105, 2011W T2 Sections 204, 205, 206, 211.
1. a) Let The University of British Columbia Midterm 1 Solutions - February 3, 2012 Mathematics 105, 2011W T2 Sections 204, 205, 206, 211 fx, y) = x siny). If the value of x, y) changes from 0, π) to 0.1,
More informationMath 10C Practice Final Solutions
Math 1C Practice Final Solutions March 9, 216 1. (6 points) Let f(x, y) x 3 y + 12x 2 8y. (a) Find all critical points of f. SOLUTION: f x 3x 2 y + 24x 3x(xy + 8) x,xy 8 y 8 x f y x 3 8 x 3 8 x 3 8 2 So
More informationMath Maximum and Minimum Values, I
Math 213 - Maximum and Minimum Values, I Peter A. Perry University of Kentucky October 8, 218 Homework Re-read section 14.7, pp. 959 965; read carefully pp. 965 967 Begin homework on section 14.7, problems
More informationPartial Derivatives Formulas. KristaKingMath.com
Partial Derivatives Formulas KristaKingMath.com Domain and range of a multivariable function A function f of two variables is a rule that assigns to each ordered pair of real numbers (x, y) in a set D
More informationHOMEWORK 7 SOLUTIONS
HOMEWORK 7 SOLUTIONS MA11: ADVANCED CALCULUS, HILARY 17 (1) Using the method of Lagrange multipliers, find the largest and smallest values of the function f(x, y) xy on the ellipse x + y 1. Solution: The
More information(to be used later). We have A =2,B = 0, and C =4,soB 2 <AC and A>0 so the critical point is a local minimum
Math 8 Show Your Work! Page of 6. (a) Find and classify any critical points of f(x, y) =x 2 +x+2y 2 in the region x 2 +y 2
More informationLet f(x) = x, but the domain of f is the interval 0 x 1. Note
I.g Maximum and Minimum. Lagrange Multipliers Recall: Suppose we are given y = f(x). We recall that the maximum/minimum points occur at the following points: (1) where f = 0; (2) where f does not exist;
More informationOptimization: Problem Set Solutions
Optimization: Problem Set Solutions Annalisa Molino University of Rome Tor Vergata annalisa.molino@uniroma2.it Fall 20 Compute the maxima minima or saddle points of the following functions a. f(x y) =
More informationPractice Midterm 2 Math 2153
Practice Midterm 2 Math 23. Decide if the following statements are TRUE or FALSE and circle your answer. You do NOT need to justify your answers. (a) ( point) If both partial derivatives f x and f y exist
More informationLESSON 21: DIFFERENTIALS OF MULTIVARIABLE FUNCTIONS OCTOBER 18, 2017
LESSON 21: DIFFERENTIALS OF MULTIVARIABLE FUNCTIONS OCTOBER 18, 2017 Today we do a quick review of differentials for functions of a single variable and then discuss how to extend this notion to functions
More informationz 2 = 1 4 (x 2) + 1 (y 6)
MA 5 Fall 007 Exam # Review Solutions. Consider the function fx, y y x. a Sketch the domain of f. For the domain, need y x 0, i.e., y x. - - - 0 0 - - - b Sketch the level curves fx, y k for k 0,,,. The
More informationMAT 122 Homework 7 Solutions
MAT 1 Homework 7 Solutions Section 3.3, Problem 4 For the function w = (t + 1) 100, we take the inside function to be z = t + 1 and the outside function to be z 100. The derivative of the inside function
More informationMath 210, Final Exam, Spring 2012 Problem 1 Solution. (a) Find an equation of the plane passing through the tips of u, v, and w.
Math, Final Exam, Spring Problem Solution. Consider three position vectors (tails are the origin): u,, v 4,, w,, (a) Find an equation of the plane passing through the tips of u, v, and w. (b) Find an equation
More informationDo Now 5 Minutes. Topic Scientific Notation. State how many significant figures are in each of the following numbers. How do you know?
Do Now 5 Minutes Topic Scientific Notation State how many significant figures are in each of the following numbers. How do you know? 1,400. 0.000 021 5 0.000 000 000 874 1 140,000,000,000,000 673,000,000,000
More information14.7: Maxima and Minima
14.7: Maxima and Minima Marius Ionescu October 29, 2012 Marius Ionescu () 14.7: Maxima and Minima October 29, 2012 1 / 13 Local Maximum and Local Minimum Denition Marius Ionescu () 14.7: Maxima and Minima
More informationFinal Exam Study Guide
Final Exam Study Guide Final Exam Coverage: Sections 10.1-10.2, 10.4-10.5, 10.7, 11.2-11.4, 12.1-12.6, 13.1-13.2, 13.4-13.5, and 14.1 Sections/topics NOT on the exam: Sections 10.3 (Continuity, it definition
More informationMath 1314 Lesson 24 Maxima and Minima of Functions of Several Variables
Math 1314 Lesson 24 Maxima and Minima of Functions of Several Variables We learned to find the maxima and minima of a function of a single variable earlier in the course We had a second derivative test
More informationPartial Derivatives. w = f(x, y, z).
Partial Derivatives 1 Functions of Several Variables So far we have focused our attention of functions of one variable. These functions model situations in which a variable depends on another independent
More informationDate Lesson Text TOPIC Homework. Verifying Equations WS 3.1. Solving Multi-Step Equations WS 3.3. Solving Equations with Rationals WS 3.
UNIT 3 EQUATIONS Date Lesson Text TOPIC Homework 3.1 4.1 Verifying Equations WS 3.1 3.2 4.2 Solving Simple Equations Pg. 193 # 6 8, 12 Use any method you wish. Verify answers to # 12 3.3 4.3 Solving Multi-Step
More informationName: Instructor: Lecture time: TA: Section time:
Math 2220 Prelim 1 February 17, 2009 Name: Instructor: Lecture time: TA: Section time: INSTRUCTIONS READ THIS NOW This test has 6 problems on 7 pages worth a total of 100 points. Look over your test package
More informationBob Brown Math 251 Calculus 1 Chapter 4, Section 1 Completed 1 CCBC Dundalk
Bob Brown Math 251 Calculus 1 Chapter 4, Section 1 Completed 1 Absolute (or Global) Minima and Maxima Def.: Let x = c be a number in the domain of a function f. f has an absolute (or, global ) minimum
More informationFINAL REVIEW FALL 2017
FINAL REVIEW FALL 7 Solutions to the following problems are found in the notes on my website. Lesson & : Integration by Substitution Ex. Evaluate 3x (x 3 + 6) 6 dx. Ex. Evaluate dt. + 4t Ex 3. Evaluate
More informationMATH Max-min Theory Fall 2016
MATH 20550 Max-min Theory Fall 2016 1. Definitions and main theorems Max-min theory starts with a function f of a vector variable x and a subset D of the domain of f. So far when we have worked with functions
More informationMath 222 Spring 2013 Exam 3 Review Problem Answers
. (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w
More informationMATH 105: PRACTICE PROBLEMS FOR CHAPTER 3: SPRING 2010
MATH 105: PRACTICE PROBLEMS FOR CHAPTER 3: SPRING 010 INSTRUCTOR: STEVEN MILLER (SJM1@WILLIAMS.EDU Question 1 : Compute the partial derivatives of order 1 order for: (1 f(x, y, z = e x+y cos(x sin(y. Solution:
More informationPractice problems for Exam 1. a b = (2) 2 + (4) 2 + ( 3) 2 = 29
Practice problems for Exam.. Given a = and b =. Find the area of the parallelogram with adjacent sides a and b. A = a b a ı j k b = = ı j + k = ı + 4 j 3 k Thus, A = 9. a b = () + (4) + ( 3)
More information14 Increasing and decreasing functions
14 Increasing and decreasing functions 14.1 Sketching derivatives READING Read Section 3.2 of Rogawski Reading Recall, f (a) is the gradient of the tangent line of f(x) at x = a. We can use this fact to
More informationHomework #6 Solutions
Problems Section.1: 6, 4, 40, 46 Section.:, 8, 10, 14, 18, 4, 0 Homework #6 Solutions.1.6. Determine whether the functions f (x) = cos x + sin x and g(x) = cos x sin x are linearly dependent or linearly
More information1. For each function, find all of its critical points and then classify each point as a local extremum or saddle point.
Solutions Review for Exam # Math 6. For each function, find all of its critical points and then classify each point as a local extremum or saddle point. a fx, y x + 6xy + y Solution.The gradient of f is
More informationSection 14.8 Maxima & minima of functions of two variables. Learning outcomes. After completing this section, you will inshaallah be able to
Section 14.8 Maxima & minima of functions of two variables 14.8 1 Learning outcomes After completing this section, you will inshaallah be able to 1. explain what is meant by relative maxima or relative
More informationReview for the First Midterm Exam
Review for the First Midterm Exam Thomas Morrell 5 pm, Sunday, 4 April 9 B9 Van Vleck Hall For the purpose of creating questions for this review session, I did not make an effort to make any of the numbers
More informationFinal Exam Study Guide and Practice Problems Solutions
Final Exam Stuy Guie an Practice Problems Solutions Note: These problems are just some of the types of problems that might appear on the exam. However, to fully prepare for the exam, in aition to making
More informationMath 265H: Calculus III Practice Midterm II: Fall 2014
Name: Section #: Math 65H: alculus III Practice Midterm II: Fall 14 Instructions: This exam has 7 problems. The number of points awarded for each question is indicated in the problem. Answer each question
More informationLESSON 21: DIFFERENTIALS OF MULTIVARIABLE FUNCTIONS MATH FALL 2018
LESSON 21: DIFFERENTIALS OF MULTIVARIABLE FUNCTIONS MATH 16020 FALL 2018 ELLEN WELD 1. Quick Review of Differentials Ex 1. Consider the function f(x) x. We know that f(9) 9 3, but what is f(9.1) 9.1? Obviously,
More informationMaxima and Minima of Functions
Maxima and Minima of Functions Outline of Section 4.2 of Sullivan and Miranda Calculus Sean Ellermeyer Kennesaw State University October 21, 2015 Sean Ellermeyer (Kennesaw State University) Maxima and
More informationMethod of Lagrange Multipliers
Method of Lagrange Multipliers A. Salih Department of Aerospace Engineering Indian Institute of Space Science and Technology, Thiruvananthapuram September 2013 Lagrange multiplier method is a technique
More informationLecture 9 - Increasing and Decreasing Functions, Extrema, and the First Derivative Test
Lecture 9 - Increasing and Decreasing Functions, Extrema, and the First Derivative Test 9.1 Increasing and Decreasing Functions One of our goals is to be able to solve max/min problems, especially economics
More informationUnit #24 - Lagrange Multipliers Section 15.3
Unit #24 - Lagrange Multipliers Section 1.3 Some material from Calculus, Single and MultiVariable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 200 by John Wiley & Sons, Inc. This material is
More informationMATH20132 Calculus of Several Variables. 2018
MATH20132 Calculus of Several Variables 2018 Solutions to Problems 8 Lagrange s Method 1 For x R 2 let fx = x 2 3xy +y 2 5x+5y i Find the critical values of fx in R 2, ii Findthecriticalvaluesoffxrestrictedtotheparametriccurvet
More informationFaculty of Engineering, Mathematics and Science School of Mathematics
Faculty of Engineering, Mathematics and Science School of Mathematics GROUPS Trinity Term 06 MA3: Advanced Calculus SAMPLE EXAM, Solutions DAY PLACE TIME Prof. Larry Rolen Instructions to Candidates: Attempt
More informationA. Incorrect! Replacing is not a method for solving systems of equations.
ACT Math and Science - Problem Drill 20: Systems of Equations No. 1 of 10 1. What methods were presented to solve systems of equations? (A) Graphing, replacing, and substitution. (B) Solving, replacing,
More informationMath 121 Winter 2010 Review Sheet
Math 121 Winter 2010 Review Sheet March 14, 2010 This review sheet contains a number of problems covering the material that we went over after the third midterm exam. These problems (in conjunction with
More informationSolutions of Math 53 Midterm Exam I
Solutions of Math 53 Midterm Exam I Problem 1: (1) [8 points] Draw a direction field for the given differential equation y 0 = t + y. (2) [8 points] Based on the direction field, determine the behavior
More information3.7 Constrained Optimization and Lagrange Multipliers
3.7 Constrained Optimization and Lagrange Multipliers 71 3.7 Constrained Optimization and Lagrange Multipliers Overview: Constrained optimization problems can sometimes be solved using the methods of the
More informationMATH 3B (Butler) Practice for Final (I, Solutions)
MATH 3B (Butler) Practice for Final (I, Solutions). Gabriel s horn is a mathematical object taken by rotating the curve y = x around the x-axis for x
More information3 Applications of partial differentiation
Advanced Calculus Chapter 3 Applications of partial differentiation 37 3 Applications of partial differentiation 3.1 Stationary points Higher derivatives Let U R 2 and f : U R. The partial derivatives
More informationMath 210, Final Exam, Spring 2010 Problem 1 Solution
Problem Solution. The position vector r (t) t3î+8tĵ+3t ˆk, t describes the motion of a particle. (a) Find the position at time t. (b) Find the velocity at time t. (c) Find the acceleration at time t. (d)
More informationInt Math 3 Midterm Review Handout (Modules 5-7)
Int Math 3 Midterm Review Handout (Modules 5-7) 1 Graph f(x) = x and g(x) = 1 x 4. Then describe the transformation from the graph of f(x) = x to the graph 2 of g(x) = 1 2 x 4. The transformations are
More informationLagrange Multipliers
Calculus 3 Lia Vas Lagrange Multipliers Constrained Optimization for functions of two variables. To find the maximum and minimum values of z = f(x, y), objective function, subject to a constraint g(x,
More informationMATH529 Fundamentals of Optimization Constrained Optimization I
MATH529 Fundamentals of Optimization Constrained Optimization I Marco A. Montes de Oca Mathematical Sciences, University of Delaware, USA 1 / 26 Motivating Example 2 / 26 Motivating Example min cost(b)
More informationCALCULUS: Math 21C, Fall 2010 Final Exam: Solutions. 1. [25 pts] Do the following series converge or diverge? State clearly which test you use.
CALCULUS: Math 2C, Fall 200 Final Exam: Solutions. [25 pts] Do the following series converge or diverge? State clearly which test you use. (a) (d) n(n + ) ( ) cos n n= n= (e) (b) n= n= [ cos ( ) n n (c)
More informationMath 21a Partial Derivatives Fall, 2016
Math 21a Partial Derivatives Fall, 2016 The solutions are available on our Canvas website (https://canvas.harvard.edu/courses/17064/). Choose Files > Koji s Section. What is the derivative? 1. For each
More informationLesson 59 Rolle s Theorem and the Mean Value Theorem
Lesson 59 Rolle s Theorem and the Mean Value Theorem HL Math - Calculus After this lesson, you should be able to: Understand and use Rolle s Theorem Understand and use the Mean Value Theorem 1 Rolle s
More informationFind the absolute maximum and the absolute minimum of the function. f(x, y) = 3x + 4y. g(x, y) = x 2 + y 2 = 25.
MATLAB assignment Math 222, Fall 2011 ROOT FINDING AND OPTIMIZATION!!! Part I: The Mathematical Prelude So the goal of this part of the assignment is to approximate solutions to nonlinear systems of equations
More informationSOLUTIONS to Problems for Review Chapter 15 McCallum, Hughes, Gleason, et al. ISBN by Vladimir A. Dobrushkin
SOLUTIONS to Problems for Review Chapter 1 McCallum, Hughes, Gleason, et al. ISBN 978-0470-118-9 by Vladimir A. Dobrushkin For Exercises 1, find the critical points of the given function and classify them
More information3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:
3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable
More informationThis exam will be over material covered in class from Monday 14 February through Tuesday 8 March, corresponding to sections in the text.
Math 275, section 002 (Ultman) Spring 2011 MIDTERM 2 REVIEW The second midterm will be held in class (1:40 2:30pm) on Friday 11 March. You will be allowed one half of one side of an 8.5 11 sheet of paper
More informationLesson 3: Solving Equations A Balancing Act
Opening Exercise Let s look back at the puzzle in Lesson 1 with the t-shape and the 100-chart. Jennie came up with a sum of 380 and through the lesson we found that the expression to represent the sum
More informationMA 123 (Calculus I) Lecture 13: October 19, 2017 Section A2. Professor Jennifer Balakrishnan,
Professor Jennifer Balakrishnan, jbala@bu.edu What is on today 1 Maxima and minima 1 1.1 Applications.................................... 1 2 What derivatives tell us 2 2.1 Increasing and decreasing functions.......................
More informationChapter 6: Rational Expr., Eq., and Functions Lecture notes Math 1010
Section 6.1: Rational Expressions and Functions Definition of a rational expression Let u and v be polynomials. The algebraic expression u v is a rational expression. The domain of this rational expression
More informationExam 2 Solutions October 12, 2006
Math 44 Fall 006 Sections and P. Achar Exam Solutions October, 006 Total points: 00 Time limit: 80 minutes No calculators, books, notes, or other aids are permitted. You must show your work and justify
More informationLecture 9: Implicit function theorem, constrained extrema and Lagrange multipliers
Lecture 9: Implicit function theorem, constrained extrema and Lagrange multipliers Rafikul Alam Department of Mathematics IIT Guwahati What does the Implicit function theorem say? Let F : R 2 R be C 1.
More informationHigher order derivative
2 Î 3 á Higher order derivative Î 1 Å Iterated partial derivative Iterated partial derivative Suppose f has f/ x, f/ y and ( f ) = 2 f x x x 2 ( f ) = 2 f x y x y ( f ) = 2 f y x y x ( f ) = 2 f y y y
More informationPolynomial functions right- and left-hand behavior (end behavior):
Lesson 2.2 Polynomial Functions For each function: a.) Graph the function on your calculator Find an appropriate window. Draw a sketch of the graph on your paper and indicate your window. b.) Identify
More informationWhich one of the following is the solution to the equation? 1) 4(x - 2) + 6 = 2x ) A) x = 5 B) x = -6 C) x = -5 D) x = 6
Review for Final Exam Math 124A (Flatley) Name Which one of the following is the solution to the equation? 1) 4(x - 2) + 6 = 2x - 14 1) A) x = 5 B) x = -6 C) x = -5 D) x = 6 Solve the linear equation.
More information1.1 Basic Algebra. 1.2 Equations and Inequalities. 1.3 Systems of Equations
1. Algebra 1.1 Basic Algebra 1.2 Equations and Inequalities 1.3 Systems of Equations 1.1 Basic Algebra 1.1.1 Algebraic Operations 1.1.2 Factoring and Expanding Polynomials 1.1.3 Introduction to Exponentials
More informationMath1a Set 1 Solutions
Math1a Set 1 Solutions October 15, 2018 Problem 1. (a) For all x, y, z Z we have (i) x x since x x = 0 is a multiple of 7. (ii) If x y then there is a k Z such that x y = 7k. So, y x = (x y) = 7k is also
More informationMath 180, Exam 2, Practice Fall 2009 Problem 1 Solution. f(x) = arcsin(2x + 1) = sin 1 (3x + 1), lnx
Math 80, Exam, Practice Fall 009 Problem Solution. Differentiate the functions: (do not simplify) f(x) = x ln(x + ), f(x) = xe x f(x) = arcsin(x + ) = sin (3x + ), f(x) = e3x lnx Solution: For the first
More informationLESSON EII.C EQUATIONS AND INEQUALITIES
LESSON EII.C EQUATIONS AND INEQUALITIES LESSON EII.C EQUATIONS AND INEQUALITIES 7 OVERVIEW Here s what you ll learn in this lesson: Linear a. Solving linear equations b. Solving linear inequalities Once
More informationMaxima and Minima. Marius Ionescu. November 5, Marius Ionescu () Maxima and Minima November 5, / 7
Maxima and Minima Marius Ionescu November 5, 2012 Marius Ionescu () Maxima and Minima November 5, 2012 1 / 7 Second Derivative Test Fact Suppose the second partial derivatives of f are continuous on a
More informationFirst Derivative Test
MA 2231 Lecture 22 - Concavity and Relative Extrema Wednesday, November 1, 2017 Objectives: Introduce the Second Derivative Test and its limitations. First Derivative Test When looking for relative extrema
More informationSolve for the variable by transforming equations:
Cantwell Sacred Heart of Mary High School Math Department Study Guide for the Algebra 1 (or higher) Placement Test Name: Date: School: Solve for the variable by transforming equations: 1. y + 3 = 9. 1
More informationMath 16A Second Midterm 6 Nov NAME (1 pt): Name of Neighbor to your left (1 pt): Name of Neighbor to your right (1 pt):
Math 16A Second Mierm 6 Nov 2008 NAME (1 pt): TA (1 pt): Name of Neighbor to your left (1 pt): Name of Neighbor to your right (1 pt): Instructions: This is a closed book, closed notes, closed calculator,
More information26. LECTURE 26. Objectives
6. LECTURE 6 Objectives I understand the idea behind the Method of Lagrange Multipliers. I can use the method of Lagrange Multipliers to maximize a multivariate function subject to a constraint. Suppose
More informationMath-2. Lesson 1-2 Solving Single-Unknown Linear Equations
Math-2 Lesson 1-2 Solving Single-Unknown Linear Equations Linear Equation: an equation where all of the letters (either variables or unknown values) have NO EXPONENTS. 4x 2 = 6 2x + 3y = 6 Previous Vocabulary
More informationMath 234 Final Exam (with answers) Spring 2017
Math 234 Final Exam (with answers) pring 217 1. onsider the points A = (1, 2, 3), B = (1, 2, 2), and = (2, 1, 4). (a) [6 points] Find the area of the triangle formed by A, B, and. olution: One way to solve
More informationSection 3.3 Maximum and Minimum Values
Section 3.3 Maximum and Minimum Values Definition For a function f defined on a set S of real numbers and a number c in S. A) f(c) is called the absolute maximum of f on S if f(c) f(x) for all x in S.
More informationMath 0312 EXAM 2 Review Questions
Name Decide whether the ordered pair is a solution of the given system. 1. 4x + y = 2 2x + 4y = -20 ; (2, -6) Solve the system by graphing. 2. x - y = 6 x + y = 16 Solve the system by substitution. If
More informationEdexcel New GCE A Level Maths workbook Solving Linear and Quadratic Simultaneous Equations.
Edexcel New GCE A Level Maths workbook Solving Linear and Quadratic Simultaneous Equations. Edited by: K V Kumaran kumarmaths.weebly.com 1 Solving linear simultaneous equations using the elimination method
More information2x (x 2 + y 2 + 1) 2 2y. (x 2 + y 2 + 1) 4. 4xy. (1, 1)(x 1) + (1, 1)(y + 1) (1, 1)(x 1)(y + 1) 81 x y y + 7.
Homework 8 Solutions, November 007. (1 We calculate some derivatives: f x = f y = x (x + y + 1 y (x + y + 1 x = (x + y + 1 4x (x + y + 1 4 y = (x + y + 1 4y (x + y + 1 4 x y = 4xy (x + y + 1 4 Substituting
More informationChapter 6. Systems of Equations and Inequalities
Chapter 6 Systems of Equations and Inequalities 6.1 Solve Linear Systems by Graphing I can graph and solve systems of linear equations. CC.9-12.A.CED.2, CC.9-12.A.CED.3, CC.9-12.A.REI.6 What is a system
More informationMath 2204 Multivariable Calculus Chapter 14: Partial Derivatives Sec. 14.7: Maximum and Minimum Values
Math 2204 Multivariable Calculus Chapter 14: Partial Derivatives Sec. 14.7: Maximum and Minimum Values I. Review from 1225 A. Definitions 1. Local Extreme Values (Relative) a. A function f has a local
More informationNeed help? Try or 4.1 Practice Problems
Day Date Assignment (Due the next class meeting) Friday 9/29/17 (A) Monday 10/9/17 (B) 4.1 Operations with polynomials Tuesday 10/10/17 (A) Wednesday 10/11/17 (B) 4.2 Factoring and solving completely Thursday
More informationMath Boot Camp Functions and Algebra
Fall 017 Math Boot Camp Functions and Algebra FUNCTIONS Much of mathematics relies on functions, the pairing (relation) of one object (typically a real number) with another object (typically a real number).
More informationLAGRANGE MULTIPLIERS
LAGRANGE MULTIPLIERS MATH 195, SECTION 59 (VIPUL NAIK) Corresponding material in the book: Section 14.8 What students should definitely get: The Lagrange multiplier condition (one constraint, two constraints
More informationMath 263 Assignment #4 Solutions. 0 = f 1 (x,y,z) = 2x 1 0 = f 2 (x,y,z) = z 2 0 = f 3 (x,y,z) = y 1
Math 263 Assignment #4 Solutions 1. Find and classify the critical points of each of the following functions: (a) f(x,y,z) = x 2 + yz x 2y z + 7 (c) f(x,y) = e x2 y 2 (1 e x2 ) (b) f(x,y) = (x + y) 3 (x
More information